See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/365847293
Correlation between effective cohesion and plasticity index of clay
Article in Geologica Balcanica · November 2022
DOI: 10.52321/GeolBalc.51.3.45
CITATIONS
READS
5
2,440
2 authors:
Boriana Tchakalova
Bulgarian Academy of Sciences
Plamen Ivanov
47 PUBLICATIONS 220 CITATIONS
21 PUBLICATIONS 45 CITATIONS
SEE PROFILE
SEE PROFILE
All content following this page was uploaded by Plamen Ivanov on 02 January 2023.
The user has requested enhancement of the downloaded file.
GEOLOGICA BALCANICA 51 (3), Sofia, December 2022, pp. 45–49.
Correlation between effective cohesion and plasticity index of clay
Boriana Tchakalova, Plamen Ivanov
Geological Institute, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 24, 1113 Sofia, Bulgaria;
e-mails: boriana@geology.bas.bg; plivanov62@geology.bas.bg
(Received: 01 November 2022; accepted in revised form: 28 November 2022)
Abstract. Correlations of engineering properties are a useful tool in geotechnical engineering practice. This
paper aims to provide a correlation between the effective cohesion and plasticity index for natural, undisturbed clay soils from the Kozloduy area (NW Bulgaria), based on the results from laboratory tests. It has
been demonstrated that there is a strong correlation between the plasticity index and the effective cohesion.
The derived regression equation can be used to estimate the effective cohesion as first approximation in preliminary design of engineering projects of Pliocene and Quaternary clays encountered in northwest Bulgaria.
Tchakalova, B., Ivanov, P. 2022. Correlation between effective cohesion and plasticity index of
clay. Geologica Balcanica 51 (3), 45–49.
Keywords: effective cohesion, plasticity index, correlation.
INTRODUCTION
The effective cohesion (cʹ) of soils is one of the most
important soil parameters that is evaluated in slope
stability and suitability for building foundations. Effective cohesion is considered as a part of the shear
strength that can be mobilized due to forces arising
at particle level and is independent from the effective stress (Lambe, 1960). As per Yong and Warkentin (1966), cʹ of soils is extremely dependent
on the interaction characteristics of the clay–water
system. Thus, cʹ is affected by the Atterberg limits
of soils.
Essentially, the Atterberg limits are controlled
by soil mineralogy, pore structure, and particle size
distribution and reflect the ability of fine-grained
soil to resist external shear loading (Seed et al.,
1966). Atterberg (1911) derives seven limits that
describe changes in the behavior of cohesive soils
at varying water content. Nowadays, in practice,
only three are in use: liquid limit (wL), plastic limit
(wP) and shrinkage limit (wS). The liquid and plastic
limits represent the plasticity characteristics of soils
and are essential in the classification of fine-grained
soil. They are also used to calculate the plasticity
index (IP), which could be correlated with many soil
properties.
Correlations between the index properties parameters and the strength and deformation properties of cohesive soils are widely employed in geotechnical engineering practices as first approximation of the soil characteristics in the preliminary design of geotechnical structures, and later as a mean
to validate the results of laboratory tests (Sørensen
and Okkels, 2013).
Beneficial empirical equations associated with
various soil properties correlated with Atterberg
limits have been provided by many researchers,
such as Fener et al. (2005), Dolinar and Trauner
(2007), Mehta and Sachan (2017), Spagnoli and
Shimobe (2020), and others.
Obtaining of the Atterberg limits in a laboratory
setting is relatively simple to perform, quick, and
inexpensive compared to tests for the determination
© БАН, Геологически институт „Акад. Страшимир Димитров“, 2021
https://doi.org/10.52321/GeolBalc.51.3.45
www.geologica-balcanica.eu
Boriana Tchakalova, Plamen Ivanov
of soil strength parameters such as cʹ. Also, obtaining cʹ requires undisturbed test samples, the derivation of which is time-consuming and costly. In that
connection, this study was carried out to predict the
drain cʹ of fine-grained soils from their IP, which
will be useful for the preliminary analysis of an engineering project. The Atterberg limits and cohesion of 33 soil samples were determined. Based on
statistical methods, an empirical equation, for the
prediction of cʹ based on IP, has been obtained for
practical use.
MATERIALS AND METHODS
The soil tests were performed on 33 undisturbed
soil samples of Pliocene and Quaternary clays from
the Kozloduy area (NW Bulgaria). The soil samples were collected from borehole cores at different depths, mainly from 10 m to 25 m below the
surface. Grain size distribution, particle density,
plasticity limits, and cohesion were determined according to BDS EN ISO/ TS 17892. Classification
of soil samples was performed according to the European Soil Classification System (ESCS).
The Atterberg limits were defined according to
BDS EN ISO/ TS 17892–12:2018. The tests were
performed at a room temperature of 20 °C by the
same operator in order to reduce the possibility of
human error. The liquid limit was obtained as recommended in the clause 5.4 of BDS EN ISO/ TS
17892-12:2018 with Casagrande apparatus with a
hard base percussion cup.
The effective cohesion was derived at saturated
consolidated-drained conditions from direct shear
test. The shear resistance envelope was obtained
by three shear tests, each using a different effective normal stress (100 kPa, 200 kPa, and 300 kPa),
performed on specimens from the same soil sample.
The least squares method was used to obtain the
corresponding cʹ values.
A variety of statistics was applied to explore the
relationship between cʹ (response variable) and IP
(predictor variable). The curve fitting procedure
was used for estimation of the regression model. In
regression analysis, curve fitting is the process of
specifying the model that provides the best fit to the
specific curve set of data.
Regression analyses were performed and singlefactor models were obtained, using the equations as
described in Table 1, where x is a predictor variable,
denotes the predicted value of the response variable
y̌ for a given x; b0, b1, and b2 – coefficients of the
independent variable; and e is Euler’s constant.
46
Table 1
Regression models
Regression Model
Equation
Simple linear
y̌ = b0 + b1x
Logarithmic
y̌ = b0 + [b1 * ln(x)]
Quadratic
y̌ = b0 + b1x + b2x2
Exponential
y̌ = b0 * e (b1 * x)
RESULTS AND DISCUSSION
The tested soils cover a wide range of classifications. Based on ESCS, samples are classified as
follows: low plasticity clay (ClL) – 13 specimens;
medium plasticity clay (ClM) – seven specimens;
high plasticity clay (ClH) – six specimens; and
very high plasticity clay (ClV) – seven specimens
(Fig. 1). Summary of the obtained results for IP and
cʹ is presented in Table 2.
Correlation and regression analysis were conducted with IP as an independent variable, and cʹ as
a dependent variable. The regression models of cʹ
and IP are presented in a graphic form in Fig. 2. The
first procedure was estimation of the strength of the
relationships between variables by Pearson’s correlation test and the F-test related to it. The F-test
was targeted to the significance of entire regression
models and to detect if the independent variable
could be used to predict the models.
Fig. 1. Tested samples shown on plasticity diagram for the
ESCS classification.
Correlation between effective cohesion and plasticity index of clay
Table 2
Summary of obtained results for IP and cʹ
Table 3
Pearson correlation and F-test results
Number of
specimens
Range of
IP, %
Range of cʹ,
kPa
Regression
model
R
R2
ClL
13
11.8–14.6
14.8–23.5
ClM
7
13.8–30.5
19.3–33.4
Exponential
Linear
0.953
0.982
0.908
0.964
F-test
F
p-value
304.745
0.000
830.572
0.000
ClH
6
32.0–46.3
36.4–53.0
Logarithmic
0.962
0.925
381.482
0.000
ClV
7
44.2–78.2
47.7–76.0
Quadratic
Power
0.983
0.968
0.966
0.938
420.450
466.874
0.000
0.000
Soil type
Fig. 2. Regression models of cʹ and IP.
Table 3 provides the results from these tests.
Pearson’s correlation coefficient (R) gives information about the magnitude of the correlation, as well
as the direction of the relationship. The obtained R
values range from 0.953 to 0.983. This indicates
that there is a very strong and positive correlation
between cʹ and IP.
The coefficient of determination (R2) was also
obtained, as it provides information of how good a
model fits the data. The derived values for R² are
closer to 1 (Table 3), signifying that the regression
models cover most of the variance of values for the
dependent variable (from 90.8% to 96.6%); the total
variation of cʹ was explained by IP and just about
9.2% of it are unclear.
The exponential model has the lowest R2 value
(0.908), but it still had a strong relationship. The
quadratic model has the highest R2 value, but it is di-
rectly related to adding predictor variable and could
not be accepted as the best-fitted model.
The results of the F-test show that the regression
modes are statistically significant (p value <0.05;
Table 3). This is sufficient evidence to conclude
that IP will provide a better fit than a model with
zero independent variable, and could be used to predict the models.
The next step of regression analysis is performing the T-test. The T-statistic measures the statistical significance of the coefficients of the independent variable in explaining the dependent variable y.
Generally, any t value greater than +2 or less than –2
is acceptable. The higher the t value, the greater the
confidence we have in the coefficient as a predictor. Low t values are indications of low reliability of
the predictive power of that coefficient (Draper and
Smith, 1998).
The outcomes of the T test analyses are summarized in Table 4. All models passed the test with the
exception of the Quadratic one. The coefficient b2 of
the model is insignificant and it has been rejected.
The derived regression models, which passed
all statistical tests and their mean absolute errors
(MAE), are presented in Table 5. In judging the
efficiency of the regression models, the lower
MAE is the better model. A MAE of zero means
a perfect model. An advantage of MAE is the fact
that its score is in the same units as the variable
of interest.
Тhe exponential and the logarithmic regression
equations show the highest MAE with values of
3.75 kPa and 3.59 kPa, respectively. The MAE of
the regression for the linear model (2.50 kPa) is
almost as low as the MAE for the power model
(2.57 kPa). The difference between them is so small
that we could use either one for the best-fitted model
if we did not take into account the R2 value.
Regression model selection criteria used in the
current paper are the highest R2 value, least MAE
and statistical significance of the model and its
47
Boriana Tchakalova, Plamen Ivanov
Table 4
T-test results of regression models
T-test
Regression
model
b0
t
p-value
b1
t
p-value
Exponential
14.613
21.011
0.000
0.022
17.457
0.000
Linear
8.476
8.487
0.000
0.776
28.820
0.000
Logarithmic
–46.186
–11.205
0.000
24.254
19.532
0.000
Quadratic
6.520
3.322
0.002
0.915
7.439
0.000
Power
2.767
21.607
0.000
0.727
21.607
0.000
b2
t
p-value
–0.002
–1.156
0.257
Table 5
Regression model equations
Regression
model
Equation
Adjusted
R2
MAE,%
Exponential
cʹ = 14.613 * e0.022IP
0.905
3.746
Linear
cʹ = 8.476 + 0.776IP
0.962
2.502
Logarithmic
cʹ = 24.254 * ln(IP) – 46.186
0.922
3.590
cʹ = 2.767 * IP0.727
0.936
2.570
Power
Table 6
Statistical data of the best fitted model
Equation
cʹ = 8.476 + 0.776*IP
R
R2
MAE
0.982
0.964
2.502
F-test
T-test
F
p value
b0
p value
b1
p value
830.572
0.000
8.476
0.000
0.776
0.000
coefficients. The best-fitted regression model for
predicting cʹ of clay soils from their IP according to these criteria is the linear regression model
(Table 6).
The uncertainty of the derived linear model is
calculated by a 95% confidence interval (blue band)
and 95% prediction interval (gray band; Fig. 3). The
limitation of the suggested equation is that it is applicable for the values of IP in the range of 11% to
78% and can be used just for preliminary estimation
of cʹ.
CONCLUSION
Fig. 3. Uncertainty of the derived linear model.
48
In order to provide a correlation between the effective cohesion and plasticity index for natural
undisturbed soils from NW Bulgaria, 33 clay soil
specimens were laboratory tested to determine their
index properties and effective cohesion. Correlation and regression analysis were performed with
Correlation between effective cohesion and plasticity index of clay
plasticity index as an independent variable, and effective cohesion as a dependent variable. The test
results confirm that the effective cohesion can be related to the plasticity index. It was observed that the
linear regression model gives the best equation with
a coefficient of determination of 0.964 and mean
absolute error of 2.50 kPa. The derived equation
likewise appears to be appropriate to a wide range
of clays with an IP value in the range of 11% to 78%.
The authors believe the suggested correlation would
be a useful assessment tool for the preliminary design stages.
Acknowledgements
The authors wish to thank two anonymous reviewers for their constructive reviews and suggestions,
which improved the quality of the manuscript. This
work has been carried out in the framework of the
National Science Program “Environmental Protection and Reduction of Risks of Adverse Events and
Natural Disasters”, approved by the Resolution of
the Council of Ministers No. 577/17.08.2018 and
supported by the Ministry of Education and Science (MES) of Bulgaria (Agreement No. DO1279/03.12.2021).
REFERENCES
Atterberg, A. 1911. Die Plastizität der Tone. Internationale
Mitteilungen der Bodenkunde 1, 4–37.
Draper, N.R., Smith, H. 1998. Applied Regression Analysis.
Third Edition. Wiley & Sons Inc., New York, 736 pp.,
https://doi.org/10.1002/9781118625590.
Dolinar B, Trauner L. 2007. The impact of structure on the
undrained shear strength of cohesive soils. Engineering Geology 92 (1–2), 88–96, https://doi.org/10.1016/j.
enggeo.2007.04.003.
Fener, M., Kahraman, S., Bay, Y., Gunaydin, O. 2005. Correlations between P-wave velocity and Atterberg limits
of cohesive soils. Canadian Geotechnical Journal 42 (2),
673–677, https://doi.org/10.1139/t04-102.
Frost, J. 2020. Regression Analysis: An Intuitive Guide for Using and Interpreting Linear Models. Statistics by Jim Publishing, State College, Pennsylvania, 358 pp.
Lambe, T.W. 1960. A mechanistic picture of shear strength in
clay. Proceedings of the Research Conference on Shear
Strength of Cohesive Soils, Boulder, Colorado, 555–580.
Mehta, B., Sachan, A. 2017. Effect of mineralogical properties
of expansive soil on its mechanical behavior. Geotechnical
and Geological Engineering 35, 2923– 2934, https://doi.
org/10.1007/s10706-017-0289-6.
Seed, H.B., Woodward, R.J., Lundgren, R. 1964. Fundamental
aspects of the Atterberg limits. Journal of Soil Mechanics and Foundations Division, Proceedings of ASCE 90,
75–105.
Sørensen, K.K.; Okkels, N. 2013. Correlation between drained
shear strength and plasticity index of undisturbed overconsolidated clays. Proceedings of the 18th ICSMGE, Paris,
France, 423–428.
Spagnoli, G., Shimobe, S. 2020. Statistical analysis of some correlations between compression index and Atterberg limits.
Environmental Earth Sciences 79, https://doi.org/10.1007/
s12665-020-09272-0.
Yong, R.N., Warkentin, B.P. 1966. Introduction to Soil Behavior. The MacMillan Company, New York, 451 pp.
49
View publication stats